Is there an easy way to uv map a circle to a half of a sphere?
Let's say you have half a sphere:

go to orthongraphic top view (Numpad 7, and  Numpad 5  to togle orhto or perspective)
select all faces, press U, to unwrap, and select Project from view (bounds)

The bounds option means that blender fit the UV map to fit the bounds of the UV image. Note that this will only work without distortion if the image used for UV mapping is square.